Let
k
k
be an algebraically closed field of characteristic zero, and let
R
=
k
[
x
1
,
…
,
x
n
]
R=k[x_{1},\dots ,x_{n}]
be a polynomial ring. Suppose that
I
I
is an ideal in
R
R
that may be generated by monomials. We investigate the ring of differential operators
D
(
R
/
I
)
\mathcal {D}(R/I)
on the ring
R
/
I
R/I
, and
I
R
(
I
)
\mathcal {I}_{R}(I)
, the idealiser of
I
I
in
R
R
. We show that
D
(
R
/
I
)
\mathcal {D}(R/I)
and
I
R
(
I
)
\mathcal {I}_{R}(I)
are always right Noetherian rings. If
I
I
is a square-free monomial ideal then we also identify all the two-sided ideals of
I
R
(
I
)
\mathcal {I}_{R}(I)
. To each simplicial complex
Δ
\Delta
on
V
=
{
v
1
,
…
,
v
n
}
V=\{v_{1},\dots ,v_{n}\}
there is a corresponding square-free monomial ideal
I
Δ
I_{\Delta }
, and the Stanley-Reisner ring associated to
Δ
\Delta
is defined to be
k
[
Δ
]
=
R
/
I
Δ
k[\Delta ]=R/I_{\Delta }
. We find necessary and sufficient conditions on
Δ
\Delta
for
D
(
k
[
Δ
]
)
\mathcal {D}(k[\Delta ])
to be left Noetherian.